CN107765252A - A kind of non-temporal synchronous underwater positioning system and method based on broadcast polls - Google Patents

A kind of non-temporal synchronous underwater positioning system and method based on broadcast polls Download PDF

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CN107765252A
CN107765252A CN201710995457.4A CN201710995457A CN107765252A CN 107765252 A CN107765252 A CN 107765252A CN 201710995457 A CN201710995457 A CN 201710995457A CN 107765252 A CN107765252 A CN 107765252A
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positioning
polling
msup
msub
mrow
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李鑫滨
徐加杰
闫晓东
孙彦龙
肖志超
张成淋
韩松
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Yanshan University
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Yanshan University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S15/00—Systems using the reflection or reradiation of acoustic waves, e.g. sonar systems
    • G01S15/02—Systems using the reflection or reradiation of acoustic waves, e.g. sonar systems using reflection of acoustic waves
    • G01S15/06—Systems determining the position data of a target
    • G01S15/46—Indirect determination of position data
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S15/00—Systems using the reflection or reradiation of acoustic waves, e.g. sonar systems
    • G01S15/02—Systems using the reflection or reradiation of acoustic waves, e.g. sonar systems using reflection of acoustic waves
    • G01S15/06—Systems determining the position data of a target
    • G01S15/46—Indirect determination of position data
    • G01S2015/465—Indirect determination of position data by Trilateration, i.e. two transducers determine separately the distance to a target, whereby with the knowledge of the baseline length, i.e. the distance between the transducers, the position data of the target is determined

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  • Physics & Mathematics (AREA)
  • Engineering & Computer Science (AREA)
  • Radar, Positioning & Navigation (AREA)
  • Remote Sensing (AREA)
  • Acoustics & Sound (AREA)
  • Computer Networks & Wireless Communication (AREA)
  • General Physics & Mathematics (AREA)
  • Measurement Of Velocity Or Position Using Acoustic Or Ultrasonic Waves (AREA)

Abstract

一种基于广播轮询的非时间同步水下定位系统及方法，系统中至少包含三个位置已知的锚节点和一个携带有应答器的定位目标。系统上电后，定位目标携带的应答器不断向外广播轮询信号，轮询信号与各个锚节点一一对应，锚节点只有在接收到属于自己的轮询信号时才做出反应，向外发出响应信号回波，否则保持静默状态。当定位目标的应答器接收到锚节点的响应回波之后，根据发送轮询信号与接收到响应信号的时间差，结合声波在水下传播速度，计算出锚节点与定位目标之间的斜距，一个定位周期结束之后，定位目标至少得到三个斜距，根据三边定位算法，结合锚节点的位置，通过最小二乘法解算出定位目标的相对位置坐标。本发明具有结构合理、使用便利等优点。

A non-time-synchronized underwater positioning system and method based on broadcast polling. The system includes at least three anchor nodes with known positions and a positioning target carrying a transponder. After the system is powered on, the transponder carried by the positioning target continuously broadcasts polling signals to the outside, and the polling signals correspond to each anchor node one by one. The anchor nodes respond only when they receive their own polling signals, Echoes the response signal, otherwise remains silent. When the transponder of the positioning target receives the response echo of the anchor node, according to the time difference between sending the polling signal and receiving the response signal, combined with the underwater propagation speed of the sound wave, the slant distance between the anchor node and the positioning target is calculated. After a positioning cycle is over, the positioning target obtains at least three slant distances. According to the trilateral positioning algorithm, combined with the position of the anchor node, the relative position coordinates of the positioning target are calculated by the least squares method. The invention has the advantages of reasonable structure, convenient use and the like.

Description

A kind of non-temporal synchronous underwater positioning system and method based on broadcast polls
Technical field
The present invention relates to Underwater Navigation field, and in particular to a kind of based on broadcast polls and non-temporal synchronous Underwater Navigation System and method.
Background technology
In recent years, with Underwater resources exploitation dynamics increasing and undersea search activity rise, Underwater Navigation gets over Paid attention to come more people, and due to the high frequency characteristics of electromagnetic wave signal so that it can not carry out effect spread under water, sound wave Method the most feasible is transmitted as signal during Underwater Navigation.Underwater Navigation is mostly based on ranging, binding site solution Calculate the relative position for drawing positioning target.In terms of time angle, the underwater positioning system based on ranging is broadly divided into two classes： 1) time synchronized ranging；2) non-temporal synchronous range finding.
In ranging process, generally combined using the underwater spread speed of sound wave and propagate duration to calculate anchor node with determining Oblique distance between the target of position.Time synchronized refers to need to keep anchor node consistent with the sequential for positioning target in ranging process Property, so as to calculate the time difference between transmitting positioning signal and receives echo-signal.The realization of existing time synchronized is It is synchronous come the deadline by stamping timestamp in water sound communication signal based on realizing underwater sound communication, however, current water Sound communication can not be realized really under water, and due to the complexity of underwater environment, underwater acoustic channel is highly prone to disturb, its The reliability of communication is very unstable.
Non-temporal synchronous range finding refer to measure anchor node and position target between oblique distance when, it is not necessary to by the two when Sequence synchronizes, but by way of repeatedly intersecting measurement and making difference, draw anchor node and position the oblique distance between target.Remove Outside this, in order to shorten locating periodically, in non-temporal synchronous alignment system, generally requiring transponder can be operated in Multiple frequency ranges, at present the technique of multi-beam transducer can not realize the preferable shape for not producing interference mutually between multiband Condition, therefore, this also brings uncertain influence to non-temporal synchronous alignment system.
Chinese Patent Application No. CN200910039154 patent of invention discloses a kind of underwater positioning system and method, can To realize the positioning of submarine target.The limitation of this method, which is embodied in, needs passage time stamp to carry out time synchronized, in the time Need to carry out information transfer using the underwater sound in synchronous process, and the complexity of underwater acoustic channel can not ensure to realize stabilized communication. Chinese Patent Application No. CN201410073287 patent of invention discloses a kind of Underwater Navigation navigation system and method.The invention Need also exist for synchronizing the time of anchor node.In addition, it is also necessary to pseudorandom spread spectrum is carried out, it is this to use multi-frequency Sound wave carry out the positioning method of underwater sound communication under water, higher requirement is proposed to transducer, the interference being subject to is also more Seriously.
The content of the invention
Present invention aims at a kind of non-temporal synchronous submarine target location tasks of completion of offer based on broadcast polls Non-temporal synchronous underwater positioning system and method.
To achieve the above object, following technical scheme is employed：Three anchor nodes are at least needed in system of the present invention With the Underwater Navigation for carrying a transponder target, each anchor node arrangement is under water and position coordinates is known, anchor section The transducer of point keeps centre frequency consistent with positioning the transducer of target response device；The transponder of positioning target uses broadcaster Formula is sent out positioning signal, and carries out the transmission of positioning signal in a manner of poll to each anchor node, and anchor node also leads to The mode replied successively is crossed to respond；Each anchor node has a time shaft of oneself, the starting point of each anchor node time shaft with First broadcast polls signal for receiving positioning target response device is starting point, is judged thereafter through the division time cycle current Whether polling signal is one's own poll positioning signal, and makes corresponding action.
Further, poll positioning signal is 10 pwm signals that frequency is fixed, and is sent when polling cycle starts.
Further, the AD acquisition ports of anchor node only one's own polling signal at hand when just open, respond AD acquisition ports are closed after end, avoid external interference from having an impact the reliability of underwater positioning system.
Localization method of the present invention, comprises the following steps：
Step 1, electric in alignment system, all anchor nodes, which are in, receives state, and AD acquisition ports are opened, and position the response of target Device starts outside broadcast polls positioning signal；
Step 2, anchor node receives positioning signal and responded；
Step 3, position target and calculate oblique distance；
Step 4, target location is positioned to resolve.
Further, in the localization method, anchor node number consecutively is A1, A2, A3, and position coordinates is respectively (x1, y1, z1)、(x2, y2, z2)、(x3, y3, z3)；Underwater Navigation target is designated as O (x, y, z)；A length of T during locating periodically；Locating periodically embeds Polling cycle；A length of t during polling cycle；Polling cycle numbering is t1, t2, t3, and anchor section numbering is corresponding with polling cycle；Open AD The acquisition port time difference is Δ t；
In step 1, the transmission cycle of positioning signal is collectively constituted by two nested cycles, and large period is locating periodically, week Phase duration T sets itself as needed；Large period is nested with minor cycle, as polling cycle, a length of t during polling cycle, that is, has T =t*i, polling signal numbering is ti, i=1,2,3, polling cycle numbering and anchor node numbering are corresponding one by one.
Further, in the step 2 of methods described：
2-1, when A1 anchor nodes receive the positioning signal in first t1 polling cycle, open timer and start to count When, and response signal is sent out, time Δt closes AD acquisition ports, and timer continues timing, waits next t1 polling cycles The arrival of signal, other anchor nodes keep silent；
2-2, when A2 anchor nodes receive first t1 polling signal, when opening timer, when timing is to t- Δ t Carve and open AD acquisition ports, prepare to receive the positioning signal of t2 polling cycles, responded after receiving positioning signal, send response Echo, then closes AD acquisition ports in t+ time Δts, and other anchor nodes keep silent；
2-3, when A3 anchor nodes receive first t1 polling signal, when opening timer, when timing to 2*t- Δs t When open AD acquisition ports, prepare to receive the positioning signals of t3 polling cycles, responded after receiving positioning signal, send response Echo, then closes AD acquisition ports in 2*t+ time Δts, and other anchor nodes keep silent.
Further, in step 3, when the transponder that the response signal of A1 anchor nodes is positioned target receives, positioning Target is designated as t1, will combine the underwater biography of sound wave according to sending positioning signal and receiving the time difference of response echo signal Speed v is broadcast, calculates positioning target and the oblique distance of A1 anchor nodes
By that analogy, positioning target will obtain the oblique distance with three anchor nodes, respectively D in a locating periodically1、D2、 D3。
Further, in step 4, the positioning target that is obtained according to the position coordinates of known three anchor nodes and measurement With the oblique distance of three anchor nodes, list position and resolve equation group：
The relative position coordinates O (x, y, z) of positioning target can be calculated according to least square method.
Compared with prior art, the invention has the advantages that：Reasonable in design, convenient and practical, calculating simply, can be completed non- The submarine target location tasks of time synchronized.
Brief description of the drawings
Fig. 1 is the operating diagram of present system.
Fig. 2 is that the involved three side positions for carrying out position resolving with positioning target according to anchor node solve in a system of the invention Calculate schematic diagram.
Embodiment
The invention will be further described below in conjunction with the accompanying drawings：
Fig. 1 show the operating diagram of the underwater positioning system based on broadcast polls, by three anchor node A1, A2, A3, and carry the positioning target composition of transponder.Anchor node arranges that the position coordinates of anchor node is respectively A1 under water (0,0,0), A2 (9,4,1), A3 (- 3, -5,1).The transponder of positioning target is sent out positioning signal, as needed positioning week Phase T is 750ms, and polling cycle t is 250ms, Δ t=50ms.Position the transponder of target and changing for each anchor node carrying Energy device frequency is 35KHz, and positioning signal is 10 35KHz pwm signal.
After system electrification, all anchor nodes, which are in, receives state, and AD acquisition ports are opened.Positioning target transponder start to Outer broadcast polls positioning signal.
When A1 anchor nodes receive t1 polling signals, opening timer timing, A1 anchor nodes give a response, are sent out ringing Induction signal, when timer timing to 50ms moment, close AD acquisition ports.
When A2 anchor nodes receive t1 polling signals, timer timing is opened, t1 polling signals are not given a response, closed AD acquisition ports are closed, until timer timing opens AD acquisition ports, wait t2 polling signals to arrive to the 200ms moment.When A2 anchor sections When point receives t2 polling signals, response signal is sent out, timer timing to 300ms moment, closes AD acquisition ports.
When A3 anchor nodes receive t1 polling signals, timer timing is opened, t1 polling signals are not given a response, closed AD acquisition ports are closed, until timer timing opens AD acquisition ports, wait t3 polling signals to arrive to the 450ms moment.When A3 anchor sections When point receives t3 polling signals, response signal is sent out, timer timing to 550ms moment, closes AD acquisition ports.
When the AD acquisition ports of anchor node are closed, anchor node is in silent status, without task action, until timer Timing is until locating periodically terminates.
When the transponder that the response signal of A1 anchor nodes is positioned target receives, positioning target is according to transmission positioning signal Time difference with receiving response signal is 8.18ms, will combine the underwater spread speed 1.5m/s of sound wave, and calculate positioning Target and the oblique distance D of A1 anchor nodes1=6.14m.
When the transponder that the response signal of A2 anchor nodes is positioned target receives, positioning target is according to transmission positioning signal Time difference with receiving response signal is 8.87ms, will combine the underwater spread speed 1.5m/s of sound wave, and calculate positioning Target and the oblique distance D of A2 anchor nodes2=6.65m.
When the transponder that the response signal of A3 anchor nodes is positioned target receives, positioning target is according to transmission positioning signal Time difference with receiving response signal is 15.178ms, will combine the underwater spread speed 1.5m/s of sound wave, and it is fixed to calculate Position target and the oblique distance D of A3 anchor nodes3=11.38m.
After each locating periodically terminates, system will return to the position of three time differences and three oblique distances for positioning target Resolving is put, it is as shown in Figure 2 that position resolves schematic diagram.According to three known anchor node position coordinateses and corresponding oblique distance, It is listed below resolving equation group：
With reference to least square method, the relative position coordinates for drawing positioning target are (3.6,4.1,2.8).
In the present invention, the answer signal of broadcast polls signal and anchor node is passed in the form of sound wave in underwater acoustic channel Defeated, the centre frequency for the transducer applied under the frequency and actual scene of its signal matches.It need not lead in position fixing process Cross the repetition of underwater sound communication or multi-frequency underwater sound signal or cross-reference carrys out deadline synchronization, during the starting of each anchor node Between since being received first t1 polling signal.
Using the present invention, it is possible to achieve the positioning of equipment such as swim underwater of underwater robot, move under water under water device, underwater sightseeing needs Ask, the stabilization of underwater positioning system is improved by the current uncertain technology of immature or reliability of use as few as possible Property and reliability.
Embodiment described above is only that the preferred embodiment of the present invention is described, not to the model of the present invention Enclose and be defined, on the premise of design spirit of the present invention is not departed from, technical side of the those of ordinary skill in the art to the present invention The various modifications and improvement that case is made, it all should fall into the protection domain of claims of the present invention determination.

Claims (8)

1.一种基于广播轮询的非时间同步水下定位系统，其特征在于：所述系统中至少需要三个锚节点和一个携带有应答器的水下定位目标，每个锚节点布置在水下且位置坐标是已知的，锚节点的换能器与定位目标应答器的换能器中心频率保持一致；定位目标的应答器采用广播方式向外发送定位信号，并且对每个锚节点以轮询的方式进行定位信号的传递，锚节点也通过依次应答的方式做出响应；各个锚节点均有自己的时间轴，每个锚节点时间轴的起点以接收到定位目标应答器的第一个广播轮询信号为起点，此后通过划分时间周期来判断当前轮询信号是否为属于自己的轮询定位信号，并做出相应的动作。1. A non-time-synchronized underwater positioning system based on broadcast polling, characterized in that: in the system, at least three anchor nodes and an underwater positioning target carrying a transponder are required, and each anchor node is arranged in the water and the position coordinates are known, the transducer of the anchor node is consistent with the center frequency of the transducer of the positioning target transponder; the transponder of the positioning target broadcasts the positioning signal outward, and sends a The polling method is used to transmit the positioning signal, and the anchor nodes also respond by responding sequentially; each anchor node has its own time axis, and the starting point of each anchor node time axis is the first time the anchor node receives the positioning target transponder. The first broadcast polling signal is used as the starting point, and then by dividing the time period to judge whether the current polling signal is its own polling positioning signal, and take corresponding actions. 2.根据权利要求1所述的一种基于广播轮询的非时间同步水下定位系统，其特征在于：轮询定位信号为频率固定的10个PWM信号，在轮询周期开始时发出。2. A non-time-synchronized underwater positioning system based on broadcast polling according to claim 1, wherein the polling positioning signal is 10 PWM signals with a fixed frequency, and is sent out at the beginning of the polling period. 3.根据权利要求1所述的一种基于广播轮询的非时间同步水下定位系统，其特征在于：锚节点的AD采集口只在属于自己的轮询信号即将到来时才打开，响应结束后关闭AD采集口。3. A kind of non-time synchronous underwater positioning system based on broadcast polling according to claim 1, characterized in that: the AD acquisition port of the anchor node is only opened when the polling signal belonging to itself is about to come, and the response ends Then close the AD collection port. 4.一种基于权利要求1所述系统的定位方法，其特征在于，所述方法包括以下步骤：4. A positioning method based on the system according to claim 1, characterized in that said method comprises the following steps: 步骤1，定位系统上电，所有锚节点处于接受状态，AD采集口打开，定位目标的应答器开始向外广播轮询定位信号；Step 1, the positioning system is powered on, all anchor nodes are in the receiving state, the AD collection port is opened, and the transponder of the positioning target starts to broadcast the polling positioning signal to the outside; 步骤2，锚节点接受定位信号做出响应；Step 2, the anchor node receives the positioning signal and responds; 步骤3，定位目标计算斜距；Step 3, locate the target and calculate the slope distance; 步骤4，定位目标位置解算。Step 4, locate the target position and calculate it. 5.根据权利要求4所述的一种基于广播轮询的非时间同步水下定位方法，其特征在于：所述定位方法中，锚节点依次编号为A1、A2、A3，位置坐标分别为(x1，y1，z1)、(x2，y2，z2)、(x3，y3，z3)；水下定位目标记为O(x，y，z)；定位周期时长为T；定位周期内嵌轮询周期；轮询周期时长为t；轮询周期编号为t1、t2、t3，锚节编号与轮询周期相对应；开AD采集口时差为Δt；5. a kind of non-time-synchronized underwater positioning method based on broadcast polling according to claim 4, is characterized in that: in described positioning method, anchor node is serially numbered as A1, A2, A3, and position coordinates are respectively ( x 1 , y 1 , z 1 ), (x 2 , y 2 , z 2 ), (x 3 , y 3 , z 3 ); the underwater positioning target is marked as O(x, y, z); the positioning cycle time is T; the positioning cycle is embedded with a polling cycle; the duration of the polling cycle is t; the number of the polling cycle is t1, t2, t3, and the number of the anchor node corresponds to the polling cycle; the time difference of opening the AD collection port is Δt; 步骤1中，定位信号的发送周期由两个嵌套周期共同组成，大周期为定位周期，周期时长T根据需要自行设定；大周期内嵌套小周期，即为轮询周期，轮询周期时长为t，即有T＝t*i，轮询信号编号为ti，i＝1,2,3，轮询周期编号与锚节点编号一一相对应。In step 1, the transmission period of the positioning signal is composed of two nested periods. The large period is the positioning period, and the duration T of the period can be set according to the needs; the nested small period within the large period is the polling period, and the polling period The duration is t, that is, T=t*i, the polling signal number is ti, i=1, 2, 3, and the polling cycle number corresponds to the anchor node number one by one. 6.根据权利要求4所述的一种基于广播轮询的非时间同步水下定位方法，其特征在于：所述方法的步骤2中：6. A kind of non-time synchronization underwater positioning method based on broadcast polling according to claim 4, characterized in that: in step 2 of the method: 2-1，当A1锚节点接收到第一个t1轮询周期内的定位信号时，打开定时器开始计时，并向外发出响应信号，Δt时刻关闭AD采集口，定时器继续计时，等待下一个t1轮询周期信号的到来，其他锚节点保持静默；2-1. When the A1 anchor node receives the positioning signal within the first t1 polling period, it starts the timer and sends a response signal to the outside. The AD acquisition port is closed at Δt, the timer continues to count, and waits for the next When a t1 polling cycle signal arrives, other anchor nodes remain silent; 2-2，当A2锚节点接收到第一个t1轮询信号时，打开计时器计时，当计时到t-Δt时刻打开AD采集口，准备接受t2轮询周期的定位信号，接收到定位信号后做出响应，发送回波，然后在t+Δt时刻关闭AD采集口，其他锚节点保持静默；2-2. When the A2 anchor node receives the first t1 polling signal, it starts the timer for timing, and when the timing reaches t-Δt, it opens the AD collection port, and prepares to receive the positioning signal of the t2 polling cycle, and receives the positioning signal Then make a response, send an echo, and then close the AD collection port at the time t+Δt, and the other anchor nodes remain silent; 2-3，当A3锚节点接收到第一个t1轮询信号时，打开计时器计时，当计时到2*t-Δt时打开AD采集口，准备接受t3轮询周期的定位信号，接收到定位信号后做出响应，发送回波，然后在2*t+Δt时刻关闭AD采集口，其他锚节点保持静默。2-3. When the A3 anchor node receives the first t1 polling signal, it starts the timer for timing, and when the timing reaches 2*t-Δt, it opens the AD collection port and prepares to receive the positioning signal of the t3 polling cycle. Make a response after locating the signal, send an echo, and then close the AD acquisition port at the time of 2*t+Δt, and the other anchor nodes remain silent. 7.根据权利要求4所述的一种基于广播轮询的非时间同步水下定位方法，其特征在于：步骤3中，当A1锚节点的响应信号被定位目标的应答器接收到时，定位目标根据发送定位信号与接收到响应信号的时间差，记为t1，将结合声波在水下的传播速度v，计算出定位目标与A1锚节点的斜距 7. A kind of non-time synchronization underwater positioning method based on broadcast polling according to claim 4, characterized in that: in step 3, when the response signal of the A1 anchor node is received by the transponder of the positioning target, the positioning According to the time difference between the target sending the positioning signal and receiving the response signal, denoted as t 1 , the slant distance between the positioning target and the A1 anchor node will be calculated by combining the propagation velocity v of the sound wave in the water 以此类推，定位目标在一个定位周期内将得到与三个锚节点的斜距，分别为D1、D2、D3。By analogy, the positioning target will obtain the slant distances from the three anchor nodes in one positioning period, which are D 1 , D 2 , and D 3 . 8.根据权利要求4所述的一种基于广播轮询的非时间同步水下定位方法，其特征在于：步骤4中，根据已知的三个锚节点的位置坐标以及测量得到的定位目标与三个锚节点的斜距，列出位置解算方程组：8. A kind of non-time-synchronized underwater positioning method based on broadcast polling according to claim 4, characterized in that: in step 4, according to the position coordinates of the three known anchor nodes and the measured positioning target and The slant distances of the three anchor nodes, listing the position solution equations: <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>-</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>y</mi> <mo>-</mo> <msub> <mi>y</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>-</mo> <msub> <mi>z</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <msup> <msub> <mi>D</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>-</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>y</mi> <mo>-</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>-</mo> <msub> <mi>z</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <msup> <msub> <mi>D</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>-</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>y</mi> <mo>-</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>z</mi> <mo>-</mo> <msub> <mi>z</mi> <mn>3</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>=</mo> <msup> <msub> <mi>D</mi> <mn>3</mn> </msub> <mn>2</mn> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><msup><mrow><mo>(</mo><mi>x</mi><mo>-</mo><msub><mi>x</mi><mn>1</mn></msub><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mi>y</mi><mo>-</mo><msub><mi>y</mi><mn>1</mn></msub><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>1</mn></msub><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><msup><msub><mi>D</mi><mn>1</mn></msub><mn>2</mn></msup></mrow></mtd></mtr><mtr><mtd><mrow><msup><mrow><mo>(</mo><mi>x</mi><mo>-</mo><msub><mi>x</mi><mn>2</mn></msub><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mi>y</mi><mo>-</mo><msub><mi>y</mi><mn>2</mn></msub><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>2</mn></msub><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><msup><msub><mi>D</mi><mn>2</mn></msub><mn>2</mn></msup></mrow></mtd></mtr><mtr><mtd><mrow><msup><mrow><mo>(</mo><mi>x</mi><mo>-</mo><msub><mi>x</mi><mn>3</mn></msub><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mi>y</mi><mo>-</mo><msub><mi>y</mi><mn>3</mn></msub><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo>(</mo><mi>z</mi><mo>-</mo><msub><mi>z</mi><mn>3</mn></msub><mo>)</mo></mrow><mn>2</mn></msup><mo>=</mo><msup><msub><mi>D</mi><mn>3</mn></msub><mn>2</mn></msup></mrow></mtd></mtr></mtable></mfenced> 根据最小二乘法即可计算得到定位目标的相对位置坐标O(x，y，z)。The relative position coordinates O(x, y, z) of the positioning target can be calculated according to the least square method.
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